Problems

A convex polygon on the plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integers coordinates that lie on the same line.

Suppose a football team scores at least one goal in each of the $20$ consecutive games. If it scores a total of $30$ goals in those $20$ games, prove that in some sequence of consecutive games it scores exactly $9$ goals total.

The prime factorization of the number $b$ is $2\times 5^2\times 7\times {13}^2\times 17$. The prime factorization of the number $c$ is $2^2\times 5\times 7^2\times 13$. Is the first number divisible by the second one? Is the product of these two numbers, $b\times c$, divisible by $49000$?

Determine all prime numbers $p$ such that $5p+1$ is also prime.

On the diagram below $AD$ is the bisector of the triangle $ABC$. The point $E$ lies on the side $AB$, with $AE=ED$. Prove that the lines $AC$ and $DE$ are parallel.

On the diagram below the line $BD$ is the bisector of the angle $\mathrm{\angle }ABC$ in the triangle $ABC$. A line through the vertex $C$ parallel to the line $BD$ intersects the continuation of the side $AB$ at the point $E$. Find the angles of the triangle $BCE$ triangle if $\mathrm{\angle }ABC={110}^{\circ }$.

How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers $54345$ and $12321$ satisfy the condition, but the numbers $23423$ and $56789$ do not.

Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set $C$ consisting of $n$ elements. How many sets can be constructed using the elements of $C$?

Does there exist a power of $3$ that ends in $0001$?

Given a natural number $n$ you are allowed to perform two operations: "double up", namely get $2n$ from $n$, and "increase by $1$", i.e. to get $n+1$ from $n$. Find the smallest amount of operations one needs to perform to get the number $n$ from $1$.